When we study the motion in the absence of magnetic field, namely, \(\mathbf {H} \equiv 0\) , \((\ref {MHD})\)  reduces to the nonhomogeneous heat conducting Navier-Stokes equations. Under compatibility conditions for the initial data, Zhong [1]} showed a Serrin-type blow-up criterion and proved global strong solutions with vacuum for small initial data, which extended the local result obtained by Cho and Kim [2]} to a global one. By employing certain time-weighted a priori estimates, they showed that strong solutions exist globally provided that some smallness condition holds true. Meanwhile, Wang et al. [3]} studied a three-dimensional initial boundary value problem with the general external force and obtained global existence of strong solutions under the assumption that the initial density is suitably small. Very recently, combining delicate energy estimates and a logarithmic interpolation inequality, the author established the global existence and uniqueness of strong solutions to the 2D Cauchy problem with large initial data and non-vacuum density at infinity.
